Evaluate limit as y approaches 0 of (cos(x-y)-cos(x+y))/y

Math
Evaluate the limit of the numerator and the limit of the denominator.
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Take the limit of the numerator and the limit of the denominator.
Evaluate the limit of the numerator.
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Take the limit of each term.
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Split the limit using the Sum of Limits Rule on the limit as approaches .
Move the limit inside the trig function because cosine is continuous.
Split the limit using the Sum of Limits Rule on the limit as approaches .
Move the limit inside the trig function because cosine is continuous.
Split the limit using the Sum of Limits Rule on the limit as approaches .
Evaluate the limits by plugging in for all occurrences of .
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Evaluate the limit of which is constant as approaches .
Evaluate the limit of by plugging in for .
Evaluate the limit of which is constant as approaches .
Evaluate the limit of by plugging in for .
Combine the opposite terms in .
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Add and .
Add and .
Subtract from .
Evaluate the limit of by plugging in for .
The expression contains a division by The expression is undefined.
Undefined
Since is of indeterminate form, apply L’Hospital’s Rule. L’Hospital’s Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.
Find the derivative of the numerator and denominator.
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Differentiate the numerator and denominator.
By the Sum Rule, the derivative of with respect to is .
Evaluate .
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Differentiate using the chain rule, which states that is where and .
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To apply the Chain Rule, set as .
The derivative of with respect to is .
Replace all occurrences of with .
By the Sum Rule, the derivative of with respect to is .
Since is constant with respect to , the derivative of with respect to is .
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Subtract from .
Multiply by .
Multiply by .
Evaluate .
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Since is constant with respect to , the derivative of with respect to is .
Differentiate using the chain rule, which states that is where and .
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To apply the Chain Rule, set as .
The derivative of with respect to is .
Replace all occurrences of with .
By the Sum Rule, the derivative of with respect to is .
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Add and .
Multiply by .
Multiply by .
Multiply by .
Differentiate using the Power Rule which states that is where .
Take the limit of each term.
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Split the limit using the Limits Quotient Rule on the limit as approaches .
Split the limit using the Sum of Limits Rule on the limit as approaches .
Move the limit inside the trig function because sine is continuous.
Split the limit using the Sum of Limits Rule on the limit as approaches .
Move the limit inside the trig function because sine is continuous.
Split the limit using the Sum of Limits Rule on the limit as approaches .
Evaluate the limits by plugging in for all occurrences of .
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Evaluate the limit of which is constant as approaches .
Evaluate the limit of by plugging in for .
Evaluate the limit of which is constant as approaches .
Evaluate the limit of by plugging in for .
Evaluate the limit of which is constant as approaches .
Simplify the answer.
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Divide by .
Combine the opposite terms in .
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Add and .
Add and .
Add and .
Evaluate limit as y approaches 0 of (cos(x-y)-cos(x+y))/y

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